We consider the minimal interacting theory of a single tower of spin j=0,2,4,... massless Fronsdal fields in flat space for which consistent covariant cubic interaction vertices are known. We address the question of constraints on possible quartic interaction vertices imposed by the condition of on-shell gauge invariance of the tree-level four-point scattering amplitudes involving three spin 0 and one spin j particle. We find that these constraints admit a local solution for quartic 000j interaction term in the action only for j=2 and j=4. We determine the non-local terms in four-vertices required in the case of spin j greater than 4 and show that these non-localities can be interpreted as a result of integrating out a tower of auxiliary ghost-like massless higher spin fields in an extended theory with a local action. We also consider the conformal off-shell extension of the Einstein theory and show that its perturbative expansion is the same as of the the non-local action resulting from integrating out the trace of the graviton field from the Einstein action. Motivated by this example, we conjecture the existence of a similar conformal off-shell extension of a massless higher spin theory that may be related to the above extended theory and may have the same infinite-dimensional symmetry as the conformal higher spin theory and thus may lead to a trivial S matrix.
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